Monomial ideal

Remarks

An equivalent form: $I$ is a monomial ideal if and only if, for every $f \in I$, each monomial that occurs in $f$ with a nonzero coefficient lies in $I$. A monomial $x^b$ lies in the monomial ideal generated by monomials $x^{a}$, $a \in A$, if and only if $b - a \in \N^n$ for some $a \in A$. By Dickson's lemma, every monomial ideal is generated by finitely many monomials, and it has a unique minimal set of monomial generators.

Example: $I = (x^2, xy)$ in $k[x, y]$ is a monomial ideal by definition. The polynomial $x^3 + x y^5$ lies in $I$, and indeed both of its monomials are multiples of $x^2$ or of $xy$.

Non-example: $I = (x + y)$ in $k[x, y]$ is not a monomial ideal. It contains $x + y$, but it does not contain the monomial $x$, since every nonzero element of $I$ is a multiple of $x + y$.